Solving Fraction And Decimal Problems A Step By Step Guide

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Understanding complex fractions can be challenging, but with the right approach, they become manageable. This section delves into simplifying the complex fraction 1/(1-1/(1-1/5)). To tackle this, we'll break it down step by step, ensuring clarity and precision.

First, focus on the innermost fraction, which is 1/5. Next, we address the expression 1 - 1/5. To subtract these, we need a common denominator. We rewrite 1 as 5/5, so the expression becomes 5/5 - 1/5, which simplifies to 4/5. Now, our original complex fraction looks like 1/(1 - 1/(4/5)).

Next, we deal with 1/(4/5). Dividing by a fraction is the same as multiplying by its reciprocal. So, 1/(4/5) becomes 1 * (5/4), which equals 5/4. Our fraction now is 1/(1 - 5/4). Again, we need a common denominator to subtract. We rewrite 1 as 4/4, making the expression 4/4 - 5/4. This results in -1/4. Thus, the fraction simplifies to 1/(-1/4).

Finally, we have 1 divided by -1/4. As before, dividing by a fraction means multiplying by its reciprocal. Therefore, 1/(-1/4) becomes 1 * (-4/1), which equals -4. Hence, the simplified value of the complex fraction 1/(1-1/(1-1/5)) is -4. This step-by-step simplification highlights the importance of tackling complex problems methodically, ensuring each operation is performed accurately to arrive at the correct solution. Understanding the underlying principles of fractions and reciprocals is crucial for mastering such problems.

Arithmetic involving fractions requires a solid understanding of common denominators and basic operations. In this section, we tackle the problem 8/7 + 8/7 - 1/2, providing a detailed explanation to ensure clarity and accuracy. The key to solving this problem lies in correctly adding and subtracting fractions by finding a common denominator.

First, let’s focus on the addition part: 8/7 + 8/7. Since the denominators are the same, we can directly add the numerators. Thus, 8/7 + 8/7 = (8 + 8)/7 = 16/7. Now, the expression becomes 16/7 - 1/2. To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 7 and 2 is 14. We will convert both fractions to have this denominator.

To convert 16/7 to a fraction with a denominator of 14, we multiply both the numerator and the denominator by 2: (16 * 2)/(7 * 2) = 32/14. Similarly, to convert 1/2 to a fraction with a denominator of 14, we multiply both the numerator and the denominator by 7: (1 * 7)/(2 * 7) = 7/14. Now, our expression is 32/14 - 7/14.

With the common denominator in place, we can subtract the numerators: 32/14 - 7/14 = (32 - 7)/14 = 25/14. Therefore, the result of 8/7 + 8/7 - 1/2 is 25/14. This fraction is already in its simplest form since 25 and 14 have no common factors other than 1. Converting it to a mixed number, we divide 25 by 14, which gives us 1 with a remainder of 11. Thus, 25/14 is equal to 1 11/14. This problem illustrates the critical steps in fraction arithmetic, emphasizing the importance of finding common denominators and simplifying results.

This section addresses a complex division problem involving mixed fractions: (2020 7/8 - 2019 3/8) / (2019 1/3 - 2018 5/6). The key to solving this problem efficiently is to first simplify each set of mixed fractions separately before performing the division. This approach makes the calculation more manageable and reduces the chances of error.

Let’s start by simplifying the first part: 2020 7/8 - 2019 3/8. We can rewrite this as (2020 - 2019) + (7/8 - 3/8). Subtracting the whole numbers gives us 1, and subtracting the fractions gives us 4/8, which simplifies to 1/2. So, the first part simplifies to 1 + 1/2, which is 3/2.

Next, we simplify the second part: 2019 1/3 - 2018 5/6. This can be rewritten as (2019 - 2018) + (1/3 - 5/6). Subtracting the whole numbers gives us 1. To subtract the fractions, we need a common denominator. The least common multiple of 3 and 6 is 6. We convert 1/3 to 2/6, so the expression becomes 2/6 - 5/6, which equals -3/6 or -1/2. Thus, the second part simplifies to 1 - 1/2, which is 1/2.

Now, we have the simplified division problem: (3/2) / (1/2). Dividing by a fraction is the same as multiplying by its reciprocal. So, (3/2) / (1/2) becomes (3/2) * (2/1). Multiplying the numerators and the denominators, we get (3 * 2) / (2 * 1) = 6/2. Simplifying this fraction gives us 3. Therefore, the solution to the complex division problem (2020 7/8 - 2019 3/8) / (2019 1/3 - 2018 5/6) is 3. This example illustrates the importance of breaking down complex problems into simpler parts and applying the rules of fraction arithmetic accurately.

Adding decimals is a fundamental arithmetic skill, and mastering it requires attention to detail, especially aligning decimal points correctly. In this section, we tackle the problem 0. 372 + 3.649 + 4.863, providing a clear, step-by-step solution to ensure accurate addition. The key to adding decimals correctly lies in aligning the decimal points vertically, which ensures that you add digits with the same place value together.

To begin, we align the decimal numbers vertically:

 0.372
 3.649
+4.863
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Now, we add the numbers column by column, starting from the rightmost column (thousandths). 2 + 9 + 3 equals 14. We write down 4 and carry over 1 to the next column (hundredths). In the hundredths column, we have 7 + 4 + 6 + 1 (carried over), which equals 18. We write down 8 and carry over 1 to the next column (tenths).

In the tenths column, we have 3 + 6 + 8 + 1 (carried over), which equals 18. We write down 8 and carry over 1 to the next column (ones). Finally, in the ones column, we have 0 + 3 + 4 + 1 (carried over), which equals 8. So, we write down 8.

Combining these results, we get 8.884. Therefore, the sum of 0.372 + 3.649 + 4.863 is 8.884. This step-by-step addition highlights the importance of aligning decimal points and carrying over digits correctly to achieve an accurate result. Decimal addition is a critical skill in everyday life, from managing finances to measuring ingredients in cooking, making it essential to master.

Summary of Solutions

In summary, we have solved the following mathematical problems:

  1. The complex fraction 1/(1-1/(1-1/5)) simplifies to -4.
  2. The arithmetic expression 8/7 + 8/7 - 1/2 equals 25/14 or 1 11/14.
  3. The complex mixed fraction division (2020 7/8 - 2019 3/8) / (2019 1/3 - 2018 5/6) results in 3.
  4. The decimal addition 0.372 + 3.649 + 4.863 sums up to 8.884.

These solutions demonstrate the application of fundamental mathematical principles in solving various types of problems, including complex fractions, fraction arithmetic, and decimal addition. Each problem highlights the importance of methodical approaches, accurate calculations, and a solid understanding of basic mathematical operations.